Subgroups — #488

Order 64.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#124	C₂	1	1	yes	maximal, characteristic
3	#103	C₂	1	2	yes	maximal, complemented
4	#102	C₂	1	2	yes	maximal
5	C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈	C₂²	1	1	yes	characteristic
7	C₂×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₈	C₂²	1	2	yes	—
9	C₄⋊C₄	—	2	2	no	—
10	C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²⋊C₄	—	2	2	no	—
12	C₄²	C₂²	1	1	yes	characteristic
13	C₂³	—	2	1	no	—
14	Q₈	—	2	2	no	—
15	D₄	—	2	2	no	—
16	C₂×C₄	—	2	1	no	—
17	C₂×C₄	—	2	2	no	—
18	C₂×C₄	D₄	1	2	yes	—
19	C₂×C₄	C₂³	1	1	yes	characteristic
20	C₈	—	2	2	no	—
21	C₂²	—	4	1	no	—
22	C₂²	—	4	2	no	—
23	C₂²	C₂×D₄	1	1	yes	characteristic
24	C₄	—	2	2	no	—
25	C₄	—	4	1	no	—
26	C₄	—	4	2	no	—
27	C₄	#41	1	2	yes	—
28	C₂	—	8	1	no	—
29	C₂	#124	1	1	yes	characteristic
30	C₂	#135	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	C₂×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.